Ships in antiquity were serviced constantly. Harbors had extensive facilities for storing and maintaining ships, most famously the ship-sheds that in military harbors housed the warships, but also including large storehouses of rigging and other necessary materials. Maintaining a fleet, private or public, was a long-term investment that was for most states prohibitively expensive because of cost--even at Athens part of the expense for the fleet had to be offloaded onto the trierarchs. We know little about the specifics of private ship maintenance, but there seems to be no reason that private shipowners didn't have similar systems of maintenance for their properties.

And what happened with the ships of large fleets built for a specific war (a la the Greek fleet that fought the Persians; did those turn in to merchant ships during peace?)

Warships were, well, weapons of war. They were not particularly seaworthy, made and designed to fight battles. They had to be beached at night and did not have room for food and water. Their crews numbered in the hundreds, most of which were rowers who had to be paid and provisioned and without whom the warship was basically useless. Merchant ships were very different, with no oars, relatively small crews, low speed, etc.--they were large cargo-haulers. It'd be like repurposing a tank into an eighteen-wheeler. Warships were frequently mothballed, since the cost of maintaining a fleet of any significant size was so prohibitively high. Generally these ships would be apparently simply abandoned--Plutarch tells an apocryphal story about a young Themistoles being shown the abandoned warships rotting on the shore. I guess it's possible that the timber could be reused, but I've never heard of that happening, and I think it'd be a bit difficult